# hw05 - MIT OpenCourseWare http/ocw.mit.edu 6.055J 2.038J...

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MIT OpenCourseWare http://ocw.mit.edu 6.055J / 2.038J The Art of Approximation in Science and Engineering Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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Z 6.055J/2.038J (Spring 2008) Homework 5 Do the following warmups and problems. Due in class on Friday, 18 April 2008 . Open universe: Collaboration, notes, and other sources of information are encouraged. However, avoid looking up answers until you solve the problem (or have tried hard). That policy helps you learn the most from the problems. Bring a photocopy to class on the due date , trade it for a solution set, and ﬁgure out or ask me about any confusing points. Your work will be graded lightly: P (made a reasonable effort), D (did not make a reasonable effort), or F (did not turn in). Warmups 1. Counting dimensionless groups How many independent dimensionless groups are there in the following sets of variables: a. size of hydrogen including relativistic effects: e 2 / 4 π± 0 , ~ , c , a 0 (Bohr radius) , m e (electron mass) . b. period of a spring–mass system in a gravitational ﬁeld: T (period) , k (spring constant) , m , x 0 (amplitude) , g . c. speed at which a free-falling object hits the ground: v , g , h (initial drop height) . d. [tricky!] weight W of an object: W , g , m . 2. Integrals by dimensions You can use dimensions to do integrals. As an example, try this integral: I ( β ) = e β x 2 dx . −∞ Which choice has correct dimensions: (a.) πβ 1 (b.) πβ 1 / 2 (c.) πβ 1 / 2 (d.) πβ 1 Hints:
| {z } 2 Homework 5 / 6.055J/2.038J: Art of approximation in science and engineering (Spring 2008) 1. The dimensions of dx are the same as the dimensions of

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hw05 - MIT OpenCourseWare http/ocw.mit.edu 6.055J 2.038J...

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